I am a nerd regarding numbers, and I have tried to figure out about looks in the game.
You got Looks Increase Points and the current value of MC can be seen in the medical version of the app from doctor. You can increase looks points by one by applying beauty cream once. But you need to wait at least 4½ hour from last time you used the cream. Otherwise, you will not get a look point. Use of night time require more than 4½ hour.
The Looks level will be determined (as the other stats) Sunday at 2400 hour as this:
Looks level 1 to 2 require 6 points
Looks level 2 to 3 require 10 points
Looks level 3 to 4 require 15 points
Looks level 4 to 5 require 20 points
So in order to level up from 2 to 3 in one week, you need to apply the cream more than once a day.
The points value is reset to zero at update.
The guide notes, that Jennifer decides actions based on looks level. In general, I have not seen girls actions changed based on looks level. However, when MC have an interaction with at least some girls, the girls increase in affection and love is influenced by Looks level. Higher Looks level cause higher increase in affection and love. When you apply hair styling (2 more in looks level), the increase will be 2 more.